Chapter 15
Landau-Ginzburg theory
We have seen in Chap. 6.1 that Phase transitions are caused most of the time by the interaction between particles, with an expectation being the Bose-Einstein condensation discussed in Sect. 14.3. One speaks of cooperative phenomena. The basis theory for understanding phase transition from a general viewpoint is the Landau-Ginzburg theory.
15.1 The Landau model
Order parameter. One assumes that there exist an order parameter x that differentiates the disordered phase from the ordered phase, such that
(1 t)β t< 1 viz T
G(T,y; x) = G (T,y) yx + ax2 + bx4 + ... (15.2) 0 − respect to the order parameter x.
181 182 CHAPTER 15. LANDAU-GINZBURG THEORY
– In order to avoid solutions with unbounded x one needs that b> 0. → ∞ – The linear coupling yx of the order parameter x to the external field y is generic. For magnetic systems it take the form HM, where H is the magnetic field and M the magnetization. Compare Sect. 3.7. – The Landau-Ginzburg functional (15.2) contains, apart from the coupling to the external field y, only even powers of the order parameter x. This is necessarily the case if the sign of x (like the direction of the magnetization for a magnetic system) does not change the thermodynamic properties of the state.
Variational minimization. The variation δG of the Landau free energy with respect to a variation δx of the order parameter is
δG = y +2 ax +4 bx3 δx, (15.3) − when T and y are constant. The stationary condition δG = 0 then leads to the relation
y +2 ax +4 bx3 = 0 . (15.4) − Local stability. The (15.4) corresponds to a local minimum if
δ2G = 2 a + 12 bx2 (δx)2 > 0 . (15.5)
We will check this condition further below.
15.1.1 Solution in the absence of an external field The solution of (15.4) for y =0 is
0 for a> 0 x = (15.6) a/(2 b) for a< 0 ( ± − where the condition for a results frompb > 0. The stability condition (15.5) then takes the form δ2G 2 a for a> 0 = (15.7) (δx)2 4 a for a< 0 − Both solutions are hence stable. Order parameter scaling. The transition from a > 0 to a < 0 describes with (15.6) a continues transition between a non-ordered to an ordered states. Expanding a = a(T ) into a Taylor series around Tc we may hence assume that T a = a0(t 1), t = , (15.8) − Tc which leads to the square-root scaling a x = 0 (1 t), t< 1 (15.9) ± 2 b − r 15.1. THE LANDAU MODEL 183
of the order parameter x below the critical temperature Tc Rescaling of parameters. Rescaling the parameters entering (15.2) we may rewrite the Landau-Ginzburg functional as 1 t 1 G(T,y; x) G (T,y) = yx + ax2 + x4, a = − , (15.10) − 0 − 4 2 where we have set a0 =1/2 and b =1/4. The solution (15.9) of the order order parameter then takes the form x = √1 t, t< 1 . (15.11) ± − Ginzburg criterion. The stability condition (15.7) breaks down when the thermal ex- pectation value (δx)2 fluctuation of the order parameter becomes too large with respect to the (quadratic)h distancei of the two solutions (15.6), viz when
(δx)2 h i 1, (δx)2 4(1 t) . (15.12) (x x−)2 ≈ h i ≈ − h + − i
The Ginzburg criterion is always violated for T Tc, viz close enough to the critical temperature, that is when thermal fluctuations induce→ transitions between the two minima x+ and x−.
15.1.2 Effect of an external field We write the rescaled solution of (15.4) as
y = P (x) (t 1) x + x3 , ≡ − which implies that we need to find the in- tersection of P (x) with the external field y.
– For T >Tc, viz for t > 1, there one solution of y = P (x) for for every y.
– For t < 1, that is for T Stability. The two extrema von P (x) are given for values of the order parameter xe1 and xe2 which are 1 t xe1,e2 = − , P (xe1,e2) = ye , ∓r 3 ± with 2 3/2 ye = (1 t) . (15.13) 3 √3 − 184 CHAPTER 15. LANDAU-GINZBURG THEORY The solutions are stable, according to (15.5), if δ2G > 0. For rescaled parameters that implies that 1 t t 1+3 x2 > 0, x2 > − = x2 (15.14) − 3 e1,2 needs to be satisfied. This implies that x1 and x3 are stable and x2 unstable. Hysteresis. The overlapping coexistence of two solutions implies a discontinuous change of the order parameter x upon increasing the external field y continuously. Upon reaching ye from below the state of the system jumps from xe1 to xe2. 15.1.3 Specific heat The differential T dG = S dT + V dP + µdN, G(T,P ; x) = G xy + a 1 x2 + bx4 − 0 − 0 T − c of the free Enthalpy G(T,P ; x) implies that the entropy is given by ∂G a ∂G ∂x S = = S 0 x2 , (15.15) − ∂T 0 − T − ∂x ∂T P c T,P P ≡ 0 where we have pointed out the stationary condition|∂G/∂x{z =} 0. The component of the en- tropy unaffected by the ordering has been denoted here by S = S (T,P )= (∂G /∂T ) . 0 0 − 0 P Entropy in the absence of an external field. Using the solution (15.6) for y = 0, 2 that is x = 0 for T >Tc and a T x2 = 0 1 for T T >Tc : S = S0(T,p), a2 T T for the entropy S. The entropy is continuous at T = Tc, but not its derivates, the telltale sign of a phase transition of second order. Compare Sect. 6.3. Specific heat jump. Using ∂S ∂S C = T , C = T 0 , P ∂T P,0 ∂T P P we obtain from (15.16) the specific heat T >Tc : CP = CP,0, 2 a0 T The specific heat is discontinuous at T = Tc, with the specific heat jump ∆CP , 2 a0 ∆CP = , (15.18) 2 bTc being easily measurable quantity. 15.2 Space-dependent order parameter Till now we assumed the order parameter x(r) to be independent of the position r. This is however correct only on the average. Interaction with a a thermal bath will generically induce excited states for all thermodynamic relevant quantities, with the order parameter being no exception. Gibbs-Duhem relation. A straightforward generalization of the Gibbs-Duhem relation (5.17) for the Gibbs enthalpy G, which we did derive for homogeneous systems, to the case of a spatially non-constant chemical potential µ = µ(T,P,y(r); x(r)) is G = µN G = d3rn(r) µ(T,P,y(r); x(r)) , (15.19) → Z where n(r) is the density of particles. Expansion of the chemical potential. We assume now that n(r) n0 is constant, entering (15.19) hence just as an overall multiplicative factor. This is≡ the case e.g. for magnetic order superconducting systems. For the expansion of the chemical potential µ as a function of the order parameter x we generalize (15.2) to t 1 1 1 µ(T,y; x) µ (T,y) yx + − x2 + x4 + ( x)2 + ... , (15.20) − 0 ∼ − 2 4 2 ∇ where the gradient term ( x)2 implies that the homogeneous state with a constant x(r) is thermodynamically∼ favored.∇ 186 CHAPTER 15. LANDAU-GINZBURG THEORY Stationarity condition. The Gibbs enthalpy G becomes stationary, with the chemical potential µ given by (15.20), when 1 δG d3r y +(t 1) x + x3 δx + d3r δ ( x)2 , (15.21) ∼ − − 2 ∇ Z Z where the variation δ ( x)2 of the gradient terms is given by ∇ 1 δ ( x)2 = x (δ x) = (δx x) 2x δx, 2 ∇ ∇ ∇ ∇ ∇ − ∇ We have used here that the variation δ and the gradient are independent operation and that independent operations commute. ∇ For the integration of the variation of the gradient term we use Gauss’ theorem, d3r (δx x) = df δx x = 0 , (15.22) ∇ ∇ ∇ Z I∂V and fixed boundary conditions, withe latter implying that value of the order parameter is given and hence fixed at the surface ∂V . The variation δx of the order parameter then vanishes on ∂V . Determining equation for the order parameter. With (15.22) we obtain δG d3r y +(t 1) x + x3 2x δx (15.23) ∼ − − −∇ Z for δG, which vanishes for arbitrary δx = δx(r) if the second-order differential equation 2 t +1 x x3 = y (15.24) ∇ − − − is fulfilled. Fluctuation in the paramagnetic state. The determining equation (15.24) for x = x(r) is both inhomogeneous, due to the term ( y) on the right-hand side, and non-linear, due the term x3. We study here the effect of− a localized external perturbation ∼ y(r) y δ(r) → 0 on the paramagnetic state, that is for T >Tc. The order parameter the vanishes for y0 = 0 and may expecting that it remains small for small y0. We may therefore neglect the term x3 (15.24), which then becomes ∼ 2 t +1 x = y δ(r) . (15.25) ∇ − − 0 Fourier transformation. Eq. (15.25) is solved by a Fourier transformation, 1 x(r) = d3k x˜(k) exp(i k r), x˜(k)= d3rx(r) exp( i k r) , (15.26) · (2 π)3 − · Z Z 15.2. SPACE-DEPENDENT ORDER PARAMETER 187 which is based on the representation 1 δ(r) = d3r exp(i k r) (2 π)3 · Z of the δ-function. Substituting (15.26) into (15.25) leads to y 1 x˜(k) = 0 . (15.27) (2 π)3 k2 + t 1 − Green’s function. For the back-transformation y exp(i k r) x(r) = 0 d3k · (15.28) (2 π)3 k2 + t 1 Z − we use spherical coordinates k = k, k r = kr cos θ , | | · such that ∞ 2 +1 3 exp(i kr) k d k 2 = 2 π dk 2 d(cos θ) exp(ik cos θ) k + t 1 k + t 1 − Z − Z0 − Z 1 4 π ∞ k sin kr = dk r k2 + t 1 Z0 − 2 π2 = exp √t 1 r . r − − The end result is y0 −r/r0(t) 1 T x(r) = e , r0(t) = , t = , (15.29) 4 πr √t 1 Tc − which hold for T > 1. The Green’s function x(r) propagates the disturbance y0 occurring at r = 0 to an arbitrary location r. ∝ Scale invariant divergence of the correlation length. The correlation length r0 entering (15.29) diverges for T c from above. The type of divergence, a powerlaw with a critical exponent ν =1/2,→ implies T scale invariance, r(t) t−ν, (c t)−ν = c−ν t−ν t−ν . (15.30) ∼ s s ∼ Rescaling the reduced temperature t by a scale factor cs does not change the functional dependence of the correlation length r0 on t = T/Tc. Scale invariance and universality. The fact that thermodynamic quantities become scale invariant close to a second order phase transition has profound consequences. It implies with (15.30) that the thermodynamics properties become independent also of the 2 2 scales of the underlying physics. Length scales like the Bohr radius 4πε0h¯ /(mee ) or energy scales like kBT do not influence the scaling behavior. 188 CHAPTER 15. LANDAU-GINZBURG THEORY UNIVERSALITY The properties of a system close to a second-order phase tran- sition become universal in the sense that the critical exponents depend only on the dimensionality of the system and of the symmetry of the order parameter. The emergence of universality is a key result of theory of phase transitions. Fluctuations in the ordered phase. For the ordered phase we write the order param- eter as x = √1 t +∆x, x3 = (1 t)3/2 +3(1 t)∆x + ... , − − − where x = √1 t is its value in the absence of an external field. The response ∆x =∆x(r) − to a perturbation y = y0δ(r) is determined via (15.24) by 2 +2(t 1) ∆x = y δ(r) . (15.31) ∇ − − 0 Solving in analogy to (15.24 ) one finds with − 1 r = [2(1 t)] ν , ν = (15.32) 0 − 2 the critical same critical exponent ν =1/2.